Information Ratio

Formula

a_t = R_port,t − R_bench,t
IR  = mean(a) / std(a, ddof=1) · sqrt(P)

Tracking error is the denominator before dividing into the mean: TE = std(a) · sqrt(P).
IR = mean(a) · sqrt(P) / TE, with TE in annualised units.
std(a) = 0 → NaN (the portfolio matched the benchmark exactly in-sample).

Relation to other ratios

• Tracking error is the annualised standard deviation alone. The information ratio puts the mean active return on top of it.
• The information coefficient is a cross-sectional correlation between a signal and later returns, not this time-series ratio. Grinold's approximation IR ≈ IC · sqrt(breadth) is a model that links them, not an identity in the data.
• A Sharpe ratio uses excess return over cash. An information ratio uses excess return over the benchmark you were hired to beat, or chose to compare with.

Pitfalls

• The benchmark has to be the one that was investable on that timing. Comparing a close-to-close portfolio with a benchmark that includes a return the portfolio could not have earned inflates the ratio.
• Alignment. Missing dates, dividends in one series only, or a different currency, all land in the active return. The function subtracts the two arrays as given.
• A high IR on a tiny active risk can be a few basis points of average difference and a tracking error that is mostly noise. Report the tracking error beside the ratio.

Python reference implementation

Requires numpy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download information-ratio.py · All snippets

import numpy as np

def information_ratio(port, bench, periods=252) -> float:
    """Annualised information ratio of port minus bench.

    Sample standard deviation. Identical series return NaN.
    """
    active = np.asarray(port, dtype=float) - np.asarray(bench, dtype=float)
    if active.ndim != 1 or active.size < 2 or not np.all(np.isfinite(active)):
        raise ValueError("need at least two finite paired returns")
    if periods <= 0:
        raise ValueError("periods must be positive")
    sd = float(active.std(ddof=1))
    if sd == 0.0:
        return float("nan")
    return float(active.mean() / sd * np.sqrt(periods))

FAQ

How is this different from the information coefficient?

The information coefficient correlates a cross-section of forecasts with a cross-section of outcomes on one date. The information ratio divides a time series of active returns by their standard deviation. Similar names, different objects.

Should the active return include fees?

If the question is what the investor kept versus the benchmark, yes. If the question is whether the holdings differed from the benchmark before costs, no. The function uses the two return series you pass in, so the choice of fees happens before the call.